A test of mode existence with applications to multimodality

Author

Minnotte, Michael C.

Date

1993

Advisor

Scott, David W.

Degree

Doctor of Philosophy

Abstract

Modes, or local maxima, are often among the most interesting features of a probability density function. Given a set of data drawn from an unknown density, it is frequently desirable to know whether or not the density is multimodal, and various procedures have been suggested for investigating the question of multimodality in the context of hypothesis testing. Available tests, however, suffer from the encumbrance of testing the entire density at once, frequently through the use of nonparametric density estimates using a single bandwidth parameter. Such a procedure puts the investigator examining a density with several modes of varying sizes at a disadvantage. A new test is proposed involving testing the reality of individual observed modes, rather than directly testing the number of modes of the density as a whole. The test statistic used is a measure of the size of the mode, the absolute integrated difference between the estimated density and the same density with the mode in question excised at the level of the higher of its two surrounding antimodes. Samples are simulated from a conservative member of the composite null hypothesis to estimate p-values within a Monte Carlo setting. Such a test can be combined with the graphical notion of a "mode tree," in which estimated mode locations are plotted over a range of kernel bandwidths. In this way, one can obtain a procedure for examining, in an adaptive fashion, not only the reality of individual modes, but also the overall number of modes of the density. A proof of consistency of the test statistic is offered, simulation results are presented, and applications to real data are illustrated.